How does hydrostatic equilibrium work?
(Visual example of hydrostatic equilibrium within the Sun provided from: woodahl.physics.iupui.edu)
Hydrostatic equilibrium, in general, occurs when compression of something due to gravity is balanced by a pressure gradient force. It occurs in anything massive enough (in the universe) that its self-gravity is strong enough to pull itself into a near spherical shape.
In stars, hydrostatic equilibrium is the state of balance it experiences through most of its lifetime. It is the "battle" between a star's own self-gravity vs its internal gas and radiation pressure.
Stars exist a majority of their lives in a phase called the main sequence. It is in this phase that stars are fusing hydrogen into helium within their cores. The mass of a star will determine the rate of its nuclear fusion and the amount of fuel that is available. When a star achieves nuclear fusion, it radiates energy through space. At the same time, internal gas pressure is also pushing outwards due to the star's substantial amount of gravity. The radiation pressure combined with the gas pressure balances the inward pull of gravity preventing the collapse of a star.
Another interesting fact is that the smaller a star is, the longer it will exist in a state of hydrostatic equilibrium.
Larger stars have more fuel, but they have to burn (fuse) it faster in order to maintain equilibrium. A faster rate of nuclear fusion in more massive stars results in the use of all their fuel in a shorter length of time. When a star loses its hydrostatic equilibrium, it will collapse under its gravity causing a supernova, or form into a white dwarf, etc (depending on its overall mass). Our Sun, for example, will spend about 10 billion years as a main sequence star before slowly expanding into a red giant (then eventually compressing into a white dwarf). A star about 10 times more massive than the Sun, on the other hand, will only last 20 million years on the main sequence.
Hydrostatic equilibrium has also become an important factor in determining whether or not some of the smaller astronomical objects in our Solar System can be identified as dwarf planets.
Like actual planets, a hydrostatic equilibrium must be achieved by smaller celestial bodies in order to be classified as a dwarf planet. This is one of the guidelines established in 2006 by the International Astronomical Union for identifying planets, dwarf planets, or small solar system bodies in the Solar System.